Determine whether the conditions $$ \angle \mathrm{Q} \cong \angle \mathrm{N}, \angle \mathrm{P} \cong \angle \mathrm{M} $$, and $$ \overline{\mathrm{PQ}} \cong \overline{\mathrm{MN}} $$ are sufficient to prove that $$ riangle \mathrm{PQR} \cong riangle \mathrm{MNO} $$ and justify your answer: Are the conditions $$ \angle \mathrm{Q} \cong \angle \mathrm{N}, \angle \mathrm{P} \cong \angle \mathrm{M} $$, and $$ \overline{\mathrm{PQ}} \cong \overline{\mathrm{MN}} $$ sufficient to prove that $$ riangle \mathrm{PQR} \cong riangle \mathrm{MNO} $$ ? No Yes Justify your answer with the congruence condition. Choose the correct answer below. AAS None ASA

Question

Determine whether the conditions $$ \angle \mathrm{Q} \cong \angle \mathrm{N}, \angle \mathrm{P} \cong \angle \mathrm{M} $$, and $$ \overline{\mathrm{PQ}} \cong \overline{\mathrm{MN}} $$ are sufficient to prove that $$ 	riangle \mathrm{PQR} \cong 	riangle \mathrm{MNO} $$ and justify your answer: Are the conditions $$ \angle \mathrm{Q} \cong \angle \mathrm{N}, \angle \mathrm{P} \cong \angle \mathrm{M} $$, and $$ \overline{\mathrm{PQ}} \cong \overline{\mathrm{MN}} $$ sufficient to prove that $$ 	riangle \mathrm{PQR} \cong 	riangle \mathrm{MNO} $$ ? No Yes Justify your answer with the congruence condition. Choose the correct answer below. AAS None ASA

UpStudy AI · Accepted Answer

To determine whether the conditions $$ \angle \mathrm{Q} \cong \angle \mathrm{N}, \angle \mathrm{P} \cong \angle \mathrm{M} $$, and $$ \overline{\mathrm{PQ}} \cong \overline{\mathrm{MN}} $$ are sufficient to prove that $$ 	riangle \mathrm{PQR} \cong 	riangle \mathrm{MNO} $$, we need to analyze the given conditions in the context of triangle congruence criteria.

1. **Identify the given conditions:**
   - $$ \angle \mathrm{Q} \cong \angle \mathrm{N} $$
   - $$ \angle \mathrm{P} \cong \angle \mathrm{M} $$
   - $$ \overline{\mathrm{PQ}} \cong \overline{\mathrm{MN}} $$

2. **Determine the congruence criteria:**
   - The conditions provided include two angles and one side. This fits the criteria for the **Angle-Angle-Side (AAS)** congruence condition, which states that if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.

3. **Conclusion:**
   - Since we have two angles and a side that is not between the angles, we can conclude that the triangles $$ 	riangle \mathrm{PQR} $$ and $$ 	riangle \mathrm{MNO} $$ are congruent by the AAS condition.

Thus, the answer is **Yes**, the conditions are sufficient to prove that $$ 	riangle \mathrm{PQR} \cong 	riangle \mathrm{MNO} $$, and the justification is based on the **AAS** congruence condition.
Yes, the conditions are sufficient to prove that $$ 	riangle \mathrm{PQR} \cong 	riangle \mathrm{MNO} $$ using the AAS congruence condition.

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